Noncommutative resolutions and CICY quotients from a non-abelian GLSM

We thank both referees for their careful reading of our paper and for very insightful comments that have helped us improve our presentation.

Concerning the second report, we have implemented all requested changes.

Concerning the first report, we have the following additional remarks to the referee's comments, labelled in the order they appear in the report:

1) We have corrected our statement of the transversality condition in the discussion around (2.11). The correct statement is that the superpotential should be such that (2.11) only has solutions where any one of the triples $x_i, y_i, z_i$ vanish. Indeed, for generic polynomials $G_i$ there can be solutions to (2.11) of the form that the referee raises. However, the D-term equations impose |x| = |y| = |z|, so those solutions of (2.11) raised by the referee do not give classical vacua of our $U(1)^3\rtimes\mathbb{Z}_{3}$ model. We have added a footnote on page 11 that concerns the phase structure of the related $U(1)^3$ model with three independent FI parameters, where such geometric branches play an important role.

2) We have added clarifying remarks on p. 14 below (2.42) and reformulated the two paragraphs below (2.44) on p. 15. See also our response on the next request of the referee for further comments.

3) We have modified and extended the last paragraph on the old p. 16 (now first paragraph on p. 18). Here are some further comments which also relate to the referee’s previous request with regards to the Coulomb branch.

We agree that conifold transitions and phenomena that usually occur at phase boundaries can also occur at limiting points in phases, already in abelian GLSMs. However, we believe that the mechanism that generates the singular behaviour is different for the model we discuss in our paper and is indeed due to non-abelian dynamics. In our model, the strongly coupled phase has a non-compact Coulomb branch everywhere in the phase. This was first observed by Hori and Tong in hep-th/0609032 (see Section 4.2 which also applies to our model). Hori and Tong then argue that, due to dynamically generated masses, this Coulomb branch is lifted everywhere except at the limiting point in the phase. The Coulomb branch is not the Coulomb branch of the GLSM but the one of the SQCD in the IR. We have made modifications in the draft to clarify this.

We have considered the example presented in Section 2.5 of arXiv:1305.5767[hep-th] and did not find any indication of the same phenomenon happening. Nor did we find that the GLSM itself has a non-compact Coulomb branch in the “exoflop”-phase. Rather, in this model, and other exotic hybrid models, the source of the singularity seems to come from the Higgs branch. We note, however, that non-regular phases have so far only been described in one-parameter non-abelian GLSMs. A generalisation to multi-parameter models and a potential connection to exoflops is an interesting open question.

At this point, we do not have an argument that could exclude the existence of noncompact Coulomb branches at limiting point in phases in abelian GLSMs which is why we have slightly weakened our statement.

4) According to Hori-Romo arXiv:1308.2438[hep-th] Section 3.1.2, the derivation of the hemisphere partition function holds for any compact Lie group $G$. We have added this to the text above (2.58). The discrete part of the gauge group enters the hemisphere partition function in two ways:

(a) The overall normalisation, which depends on the Weyl group of the gauge group. While the normalisation of the hemisphere partition function is not fixed, the choice which is suggested by the topological data of the Calabi-Yau in the $\zeta>0$ - phase has a factor 1/3 (see above eq.(2.67)) which is expected from the Weyl group. See our comment in the paragraph between (2.66) and (2.67).

(b) The brane factors: The D-branes need to be gauge invariant. The discrete symmetry forbids certain matrix factorisations which would be allowed in its absence. For example, consider the superpotential

$W = (x_1)^3+ (x_2)^3$

and the associated matrix factorisation

$Q = (x_1 + x_2)\eta + (x_1^2− x_ 1 x_2 + x_2^2) \overline{\eta} $

($\eta, \overline{\eta}$ being the usual Clifford matrices).

Such a matrix factorisation will not be invariant under a $\mathbb{Z}_3$ which acts differently on $x_1$ and $x_2$. We have slightly extended footnote 14 on p. 20 to clarify this. In our example, we believe we have not missed any further contributions. For example, we found a consistent way to match the expected topological invariants of the large volume phase of our model by computing the hemisphere partition function for the brane associated to the structure sheaf.

6) We have altered our discussion of $c_2$ here. The fact that the leading term in the genus-1 free energy for X matches $c_2(X_{def})$ was observed in reference [2], where other techniques were available to fix this leading term. We have taken this as a basis for our assumption that in our example this leading term is $c_2(X_{def})$. We have stated this, with extra discussion, above (3.26) on page 28 (in the new version).

A reference to the paper of Batyrev and van Straten was added, used below equation (1.2).

On the first paragraph of page 2, we add references [14-19] for other relevant GLSM constructions.

For the reader's convenience, we have added a diagram in Section 1.1 that collects the geometries we study, displaying their relations.

Around equation (2.11), we have modified our discussion of the transversality requirement. Also we have corrected a typo in (2.12), this does not affect our analysis or conclusions.

Above (2.24), we clarify which combination of $\sigma$-fields vanish.

After (2.25), we have modified our discussion in light of the above transversality condition, and added a footnote with further discussion in light of Referee 1's comments.

Clarifying remarks were added below (2.42) and (2.44), in light of Referee 1's comments.

We changed the wording below (3.5) to prevent confusion as to which solutions were being referred to.

Second paragraph of S3.2.2, we correct our phrasing on Calabi-Yau versus $c_1=0$.

Our discussion of $c_2$ above equation (3.26) has been altered, in line with Referee 1's comments.

Below (3.25), a reference to Namikawa-Steenbrink was added.

End of paragraph below (3.25), we added two relevant references to Braun-Kreuzer-Ovrut-Scheidegger.

At the top of page 30, we improve our phrasing concerning the Witten index.

In the second paragraph of section 4, we discuss briefly the applicability of our methods to other AESZ operators.

Below (A.20) we reference Hosono-Konishi for this gap observation.

Holomorphic ambiguities at genus 2, 3, and 4 have been included on page 49.